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Nonlinear transmission problems in boundeddomains of Rn

Klaus Pflüger aa Institut fiir Mathematik I, Freie Universität Berlin, Berlin, 14195, Germany

Version of record first published: 02 May 2007

To cite this article: Klaus Pflüger (1996): Nonlinear transmission problems in bounded domains of Rn , ApplicableAnalysis: An International Journal, 62:3-4, 391-403

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Nonlinear Transmission Problems in Bounded Domains of Rn Communicated by: E. Meister

Klaus Pfliiger

Freie Universitat Berlin Institut fiir Mathematik I

Arnimallee 3 14195 Berlin, Germany

Abstract

We study a semilinear elliptic problem in a bounded domain of lRn (n 2 2) with nonlinear transmission conditions on a submanifold of the domain. This problem is related to electromagnetic wave propagation in fibers, where such conditions appear at the interface between the core and the cladding of the fiber. We present a variational formulation of this problem and obtain a weak solution by means of the mountain pass lemma. By constructing suitable trace spaces, we show that this solution satisfies the required interface conditions in a generalized sense.

KEY WORDS: semilinear elliptic equation, nonlinear transmission condition, variational methods, electromagnetic waves in fibres.

(Received for publication July 1996)

1 Introduction

T h e a im of this paper is t o present a variational formulation for certain interface prob- lems for elliptic operators in bounded domains of Rn. Such problems occur for example in physical systems in different connected media. For a general discussion of these prob- . .

lems and examples where internal layers and interface problems occur, we refer t o the monograph of P. C. Fife [4]. For linear elliptic transmission problems on quit general two

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dimensional domains (L'polygonal topological networks"), existence and regularity ques- tions have been intensively studied (see for example [9]). On the other hand, interaction and transmission problems for wave equations on networks have been examined in [2]. In that paper also an approach to nonlinear interface problems is suggested, which uses maximal monotone operators on convex subsets of products of certain function spaces.

In this paper we study an interface problem for semilinear elliptic equations with (rather natural) nonlinear transmission conditions, which cannot be solved by monotone operator techniques. Furthermore, our conditions at the interface do not lead to convex subset,s of the associated function spaces, so that we have to use a different approach to handle these problems.

The present paper is motivated by the study of electromagnetic wave propagation in fibers. Mathematically, a fiber is a cylindrical domain of the form G = R x El, where R C an is bounded, consisting of two subdomains G1 = R1 x R (the core) and G2 = n2 x lR (the cladding) with distinct dielectrical properties. Under the physically reasonable assumption that the magnetic permeability is constant in G, the different properties of G1 and G2 are described by the electric permittivity ~ ( x ) , which is assumed to depend only on the cross section of the fiber, i. e. x E R. To describe electromagnetic waves in G, we have to consider Maxwell's equations in G :

where E is the electric field, H the magnetic field, D = E(x)E + P ( x , E ) , and P de- notes the polarization vector which may depend nonlinear on E. For example, a cubic nonlinearity of the form P ( x , E) = p(x)l El2 E is known as Kerr-effect (cf. [ 7 ] , [6]). As-

suming that nonlinear effects can only appear in the core of the fiber (i. e. p ( ~ ) = 0 for x E R,), and that E ( X ) is discontinuous at the interface n1 n az, the electric field E itself has discontinuities at this interface. Therefore, the divergence condition div D = 0 must be interpreted in the distributional sense. Denote Dl = DlG, , D2 = DIG,; taking the divergence of a discontinuous function leads to the appearance of a delta distribution concentrated on the submanifold of discontinuity. Therefore the divergence condition for D leads to the following three conditions:

where n is the normal vector at the interface pointing outside G1. Since D depends nonlinear on E , the last condition is a nonlinear transmission condition for E at the interface. From Maxwell's equations we derive another transmission condition for E : Since we assume the magnetic permeability to be constant in G, the magnetic field H and its time-derivative dtH should be continuous in all of G. Consequently, denoting El = Elcl , E2 = E I G ~ , we obtain from -rot E = dtH the condition

rot El = rot E2 on 771 n G2,

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which is a linear interface condition for E .

If we look for a periodic wave train for these nonlinear Maxwell equations with (1' inear and nonlinear) interface conditions, i. e, for solutions E( t , x, y) = U ( X ) ~ ~ ( P Y - ~ ~ ) (where x E R, y E R) , we are lead t o a stationary problem of elliptic type in the bounded domain R. Note that the nonlinear function P satisfies P ( x , E) = p(x) lU(x) 12~(z)ei(fly-wt).

In the following we shall study an elliptic model problem in a bounded domain of Rn with transmission conditions of the type described above. We shall derive a variational formulation for this problem and show that this variational problem admits a solution, which satisfies the transmission conditions in the sense of gneralized traces of Sobolev spaces. We shall restrict ourself to the situation where a given domain in Rn consists of two different subdomains, where on one of them we have given a nonlinear elliptic equation, while on the other one we have a linear elliptic operator.

To be more precise, let R C En (n 2 2) be a bounded, connected domain and 01, R2 disjoint open subdomains of 52 such that = nl u IT2 and I' = n1 i;l IT2 is such that int (I') c int (0). We denote I ' e = d R L \ T , e = 1 , 2 .

In the picture above both I'l and F2 are not empty. However, it is also possible that I'l = 0. In this case I' = dR1 and all conditions and statements on I'l below have to be dropped. We assume through- out the paper that Re, e = 1,2 are Lip- schitz domains.

For l = 1,2 we define the differential and boundary operators

where nj(J) are the components of the unit normal vector in E E I', which points outward

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with respect to R1. Then we study the following problem:

where f : S1 x lR --+ lR, y : r x lR --i lR are Caratheodory functions satisfying several growth conditions, which are specified below.

2 Notations and Main Results

Let F and a) denote the primitive functions of f and p. We assume that the following conditions are satisfied:

1" The coefficients a h r e in C1(IIt), a h = aSi, ae(x) > a > 0, and A, satisfy the ellipticity conditions Crj=, afj(x)(i(j 2 ael(12 for every ( E lRn, where cue > 0.

2" There exists a real 0 E [O, 112) such that for every x E 01, [ E I?, t # 0

Furthermore, there exists an open, nonempty set 0 c R1 and a number R > 0, such that F ( x , t ) > 0 for every x E 0, t 2 R.

kt2 3" If n 2 3, we assume limltl,, f (z , t)/lt/n-2 = lirnltl+m p(E, t ) / l t lA = 0,

lim sup f (x , t ) / t = lim sup p((, t ) / t = 0 ltl-+O ItI+JJ

uniformly in x and E. If n = 2, the exponents ( n + 2) / (n - 2 ) and n/(n - 2) may be replaced by any finite power p.

Remark. In 2", it would be sufficient to assume the inequalities to hold for It1 2 R, where R is any positive constant. But for simplicity and to avoid too much notations, we will use condition 2" in the present form.

We define the bilinear form induced by A, as

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and the energy functional by

where for functions ue E H1(Re) the terms in the boundary integral are to be understood in the sense of traces

We define CF(Qe) as the set of all C"- functions 7 with supp(7) compact in R (i. e. 7 is the restriction of a function + E C r ( R ) to the subdomain Re. Finally let 71 be the completion of

in the norm [lvll = (117~ l l ~ ~ ~ ~ , , + 1172 l(Ll(n,))112. We remark that, in view of lo, this norm 112

isequivalent tothenorm ( ~ l u l / i l + ~ ~ u 2 1 l ~ ) ,where I u ~ ~ ~ ~ , = ( u ~ , u ~ ) ~ , . Obviously71

is a Hilbert space. As a subspace of H1(R1) x H1(R2) the following embedding, resp. trace operators

are continuous for 1 5 p, r 5 2n/(n - 2), resp. 1 5 q, s 5 2(n - l ) / ( n - 2) and compact for 1 < p, r < 2n/(n - 2), resp. 1 < q, s < 2(n - l ) / ( n - 2) (if n 2 3); if n = 2 they are compact for 1 5 p, q, r , s < a. From this and from our assumptions lo-3' it follows that J is Frkchet differentiable on 71 with derivative

for every v = (vl, vz) E 71. The first result of this paper is

Theorem 1 There exists a nontrivial critical point of J in 71.

Next we show that this critical point is a weak solution of the original transmission problem where the transmission conditions are satisfied in the sense of traces in Ls(F) and HG' /~(~?) (for a definition see Section 4).

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Theorem 2 There exists a nontrivial weak solution in 3-1 of Problem (1)-(5)) where (3) is satisfied in H F " ~ ( I ' ) and (4) holds in Ls(I') , where s = 2 ( n - l ) / n if n 2 3 and s is any number 2 1 if n = 2.

The paper is organized as follows. In Section 3 we prove Theorem 1. In Section 4 we recall some facts about traces in Lipshitz domains and Section 5 is devoted to the proof of Theorem 2.

3 Existence of a Critical Point

We want to apply the Mountain-Pass lemma (cf. [ l l ] ) to show the existence of a critical point of J . Therefore we have to show the following:

(i) J satisfies the Palais-Smale condition, i. e. every sequence uk E 3-1 with ( J ( u k ) l < M < ca and J 1 ( u k ) + 0 in 7 i * has a convergent subsequence in 3-1.

(ii) There are numbers p, S > 0 , such that J ( u ) >_ S for every u E 'FI with \lull = p.

( i i i ) J ( 0 ) = 0 and there exists e E 3-1, llell > p, such that J ( e ) < 6.

We start with the following lemma.

Lemma 3 Every sequence satisfying the assumptions in (i) is bounded in 3-1.

Proof. From I ( J 1 ( u k ) , v ) l 5 ~ l l v l l for every v E 'FI and sufficiently large k , we obtain (setting v = u k )

Since J ( u k ) is bounded, we have Dow

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NONLINEAR TRANSMISSION 397

Now we set E = 1 and multiply (8) with 6 E [O, 112) from assumption 2" and add this to (9) , to obtain

Since ae are positive, we obtain

where Cr is the trace constant with respect to the norm 1 1 . ] I A 1 , i. e. 1 1 ~ 1 1 ; 2 ( ~ ) 5 CrlluII;, for every u E H1(R1). Since the transmission condition (4) is equivalent to

t (al(()ul + C ~ ( E , U ~ ) ) = t a2(()u2 on r for some t > 0 ,

we may assume without any restrictions that

Now it is easy to show that uk contains a Cauchy sequence. One has to use assumption 3", Holder's inequality and the compactness of 7-l FI Lp(R1) and 'FI - Lq(r) x LS(I'). Thus, J satisfies the Palais-Smale condition.

To prove (ii), we observe that from 3" we find to any E > 0 a constant C, > 0 such that

We claim

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Since a l ( z ) 2 cu > 0 on IT1, we can choose E 5 a12 and get

where m i n { n ( n - 2 ) 2 - 1 ) - 2 if l l u l l l ~ ~ 5 1

= { max{2n/(n - 2), 2(n - l ) / ( n - 2)) if ~ ~ u I ~ ~ A ~ > 1

Here we have used the Sobolev inequalities

This shows that for llull = p small enough, we get J ( u ) > S > 0.

It remains to show (iii). Observe that J ( 0 ) = 0 follows from our assumptions. Now let 7 be any positive function with compact support in 0 c R1 ( 0 as in 2") with 7 > 0 somewhere. From 2' we infer, if 7 2 R, F ( z , 7) > h ( ~ ) ~ ~ / ' for x E 0, where h(x) > 0. Consequently, for real t > 0, we obtain

where C2 > 0. Since 110 > 2, we obtain J ( t 7 ) -+ -co as t -+ co and for t large enough, e = t7 satisfies condition (iii).

This completes the proof of Theorem 1

4 Traces

The main difficulty in the above problem lies in the transmission conditions (3) and (4) . To control the behaviour at the boundary of a minimizing sequence, we need some trace theorems for Lipschitz domains. Following Grisvard [ 5 ] , we introduce the spaces

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and E > 0 is an arbitrary small real number. On I: we define the graph norm IIv =

l l v l l ~ ~ + IIA~vIIL~. We denote

For bounded Lipschitz domains the following Green formula holds for ue E H2(Re), ve E H,'(Re) (cf. [81)

(Aeue, ve)o - (ue, v e ) ~ , = (-1)' l ~ e u e w d r , (10)

where the sign (-l)e is due to t h e fact that the normal vector a t the boundary I' in the definition of Be points outward R1 and inward R2. As a particular case of this formula we obtain

(Aeue, ve)o = (ue, v e ) ~ , (11)

if ue E H,'(Re) satisfies Aeue E LT(Re) ( r 2 n,), and ve E C,"(Re).

Since the trace operator y : H1(Re) + H ~ / ~ ( ~ R ~ ) has a continuous right inverse and for every u E H,I(Re) we have yu = 0 on r e , it follows that there exist linear, continuous operators

Denoting ve = At($), every element ue E H,'(Re) n E,' defines a continuous linear func- tional A,, on H,'/~(I') by the formula

for sufficiently smooth functions ue.

The fact that Xu, is continuous on H ~ / ~ ( I ? ) follows from the continuity of the bilinear form (., . ) A , and from Holder's inequality: I(Aeue, ~ ~ ) ~ l < IIAeuellT llvtllrt, where l / r + l / r l = 1. Since r > n,, we claim r' < co (if n = 2) and r' 5 2n/(n- 2) (if n > 3). The boundedness of Xu, then follows from the Sobolev inbedding theorem.

Let H-'l2(I') be the dual space of H,'/'(I'). Then, in the sense of (12)) (13)) Be are continuous, linear operators

if r > n,. To handle the transmission conditions (3)) (4) we introduce the following trace spaces. Denote ye : Hi(&) -+ H ~ / ~ ( I ' ) . Then

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is a continuous, linear operator. For the closed subspace

%FIT = ((v1,vz) E 7-C I1711v1 = yzvz)

of 7-C we define

~ t / : ( r ) := 71 (HT) (= Y ~ ( 7 - l ~ ) ) and H;"~(F) = the dual of ~,$(r) .

5 Proof of Theorem 2

Let u = (u l , u2) E 3-1 be a critical point of J according to Theorem 1. From (J 'u , v) = (1

for every v E 7-C, we obtain, taking as test function some v = (vl, v2) E C r ( R 1 ) x C F ( R 2 ) , that ue are distributional solutions of

From our growth condition 3' in Section 1 we infer that

2n f (x, ui) E Lr(R1) for 1 5 r 5 - (if n > 3) and I < r < cto (if n = 2) .

n + 2

Therefore, B lu l is defined in the sense of (12), (13). (15)

Since Do is dense in 3-1, we can choose a sequence uk = (uk,l, uk,2) E DO satisfying

Jt(uk) -+ 0 , uk,e -+ ue in the norm 1 1 . and 1 1 . II.vlcnp,

Now from Bluk,l = B Z U ~ , ~ for every k we derive the following lemma.

Lemma 4 The critical point u = (ul , u2) E 7-C of J satisfies Blu l = B2u2 on F in the sense of traces in H;""(r).

Proof. From J1(uk) --+ J1(u) = 0 we can choose to given v = (vl ,vz) E 7-C and E > 0 an index k, such that for every k > k , the inequality

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holds. From (15) we infer that (Alul ,v l )o is defined for every vl E H1(Rl ) . Using formulas ( l o ) , (12), (13), we obtain

(and the corresponding formulas for uk,l, uk,2).

From our assumption 3' we infer that the convergence u k , ~ --+ ue in the norm 1 1 . ] I A , implies that f (x ,uk , l ) --t f ( x , u l ) , in LT(R1) ( r > 1) and ~ k , l -+ U I , U k , 2 -+ ~ 2 , y ( t , uk,2) +

p( t , u2) in L r ( r ) ( r > 1) . Furthermore, in Lemma 5 below we prove that (A1uk,1, v1),, -+

(Aiul , vi)o and (Azuk,2, v2)o + (Azuz, v2)o.

Now we take a test function v in inequality (16) which satisfies yvl = yv2 = $J in H' / ' ( I ' ) . Since (uk,1, U ~ J ) E DO satisfy B1uk,l = B2uk,2 on r , we obtain from ( 1 6 ) and (17) for sufficiently large k the estimate

for all v E 'H satisfying vl = v2 = II, on I?. Since we can extend every function d E 112 Ho ( r ) to a function v E 'H such that yvl = $ = yv2, we have proved Blul = B2uZ in

~;""(r ). 0

Proof. We only prove the first statement, since the second one is similar. Let w E C,"(Rl) be an arbitrary test function. For uk,l E CT(R1) we can apply the Green formula (10). Since U I E H1(Q1) is a distributional solution of Alul = f (z, u l ) E L T ( R l ) , we can use formula (11) to obtain

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The strong convergence uk,l -+ u1 in / I . / I A , implies the weak convergence (uk,l, w ) ~ , -+

(ul , w ) ~ , and therefore

Since C r ( R 1 ) LP(R1) for p = r / ( r - 1) is a dense subspace, we see that (A1uk,l - Alul , w)O -+ 0 for a dense subset of functions w in LP(Rl). NOW LP(R1) is the dual space of LT(RI) and we conclude that (Aluk,l - A1ul) - 0 weakly in LT(R1). In particular, since every function v E H1(Rl) defines a continuous linear functional on LT(R1), the assertion of the lemma follows.

Lemma 6 A critical point u E 'H of J satisfies (4) i n Ls((r, where s = 2(n - l ) / n i f n > 3 and s is a n y number 2 1 if n = 2.

Proof. From Lemma 3 we know that B lu l = B2u2. Now using (12), (13) and the fact that u l , u2 are solutions of Alul = f(x, ul ) , A2u2 = 0, resp., we claim that

for every function $ E Cr( l7 ) . Since p ( ( , u l ) + al(()ul - a2(E)u2 E LS(I') and C,M(r) is dense in the dual space of L s ( r ) , we claim y ( t , u l ) $ a l ( t )u l = a2(E)uz

The proof of Theorem 2 now follows from Lemma 4 and Lemma 6.

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[3] A. Ambrosetti, P. H. Rabinowitz. Dual variational methods i n critical point theory and applications. J . Funct. Anal.14, 349 - 381 (1973)

[4] P. C. Fife. Dynamics of internal layers and diffusive interfaces. CBMS-NSF Reg. Conf. Ser. Appl. Math. 53 (1988)

[5] P. Grisvard. Elliptic problems i n nonsmooth domains. Pitman, London (1985)

[6] A. Hasegawa. Optical solitons i n fibers. Springer, New York etc. (1989)

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[7] V. I. Karpman. Nonlinear waves in dispersive media. Pergamon Press, Oxford etc. (1975)

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